Strong 3-skew commutativity preserving maps on prime rings with involution

ABSTRACT

Let $\mathcal{R}$ be a unital prime $\ast$-ring containing a nontrivial symmetric idempotent. For $A,B\in \mathcal{R}$, the 3-skew commutator is defined by ${}_{*}{[A,B]}_3{=}_{*}[A{,}_{\ast }{[A,B]}_2]{=}_{*}[A{,}_{*}[A{,}_{*}[A,B]]]=A^{3}B-3A^{2}B{A}^{*}+3AB{(A^{*})}^2-B{(A^{*})}^3$. Let $\Phi :\mathcal{R}\to \mathcal{R}$ be a surjective map. It is shown that $\Phi$ satisfies ${}_{\ast }{[\Phi (A),\Phi (B)]}_3{=}_{\ast }{[A,B]}_3$ for all $A,B\in \mathcal{R}$ if and only if there exists $\lambda \in {\mathcal{Z}}_S(\mathcal{R})$ with ${\lambda }^4=I$ such that $\Phi (A)=\lambda A$ for all $A\in \mathcal{R}$. Where I is the unit of $\mathcal{R}$ and ${\mathcal{Z}}_S(\mathcal{R})$ is the symmetric center of $\mathcal{R}$. This result then is applied to some operator algebras.

Communicated by Igor Klep

KEYWORDS:

• 3-skew commutators
• C ${}^{*}$-algebras
• preservers
• prime rings with involution
• symmetric standard operator algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W10
• 47B49
• 47B47

Acknowledgments

The authors wish to give their thanks to the referees for their helpful comments and suggestions that make much improvement of the paper.

Additional information

Funding

This work is partially supported by Natural Science Foundation of China (12071336,12171290) and Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (20200011).